A Parallel Algorithm for Solving the 3d Schrodinger Equation

TL;DR

This paper presents a parallel algorithm using FDTD for efficiently solving the 3d Schrödinger equation, enabling large-scale quantum simulations with optimized parallelization and new eigenvalue determination methods.

Contribution

It introduces an optimized parallelization scheme, a novel method for accurate eigenvalue and wavefunction calculation, and discusses multi-resolution techniques for large lattice problems.

Findings

Compute time scales inversely with number of nodes (~N_nodes^-0.95)

Enables solving large 3d Schrödinger problems on small clusters

Provides a new symmetry-based eigenvalue determination method

Abstract

We describe a parallel algorithm for solving the time-independent 3d Schrodinger equation using the finite difference time domain (FDTD) method. We introduce an optimized parallelization scheme that reduces communication overhead between computational nodes. We demonstrate that the compute time, t, scales inversely with the number of computational nodes as t ~ N_nodes^(-0.95 +/- 0.04). This makes it possible to solve the 3d Schrodinger equation on extremely large spatial lattices using a small computing cluster. In addition, we present a new method for precisely determining the energy eigenvalues and wavefunctions of quantum states based on a symmetry constraint on the FDTD initial condition. Finally, we discuss the usage of multi-resolution techniques in order to speed up convergence on extremely large lattices.